Temporal Exponential Random Graph Models (TERGMs) for dynamic network modeling in statnet

Prerequisites

This workshop assumes basic familiarity with R, experience with network concepts, terminology and data, and experience with Exponential family Random Graph Models (ERGMs).
The statnet ERGM workshop is ideal preparation.

Software installation

Minimally, you will need to install the latest version of R (available here) and the statnet packages above to run the code presented here.
The workshops are conducted using the free version of Rstudio (available here).

To install the packages for this tutorial, open an R session and type:

install.packages('statnet')
install.packages('ndtv')
install.packages('htmlwidgets')
install.packages('latticeExtra')

The two non-statnet packages here (htmlwidgets and latticeExtra), provide some nice graphing utilities, and demonstrate one of the many benefits of working in the R software ecosystem—you’re not restricted to the features in any one package. Throughout this tutorial we will be exploiting this benefit, using functions from base R and other packages seamlessly with our statnet software.

Make sure the packages are attached:

library(statnet)
library(ndtv)
library(htmlwidgets)
library(latticeExtra)

Check package versions

sessionInfo()

R version 3.6.0 (2019-04-26)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 17134)

Matrix products: default

locale:
[1] LC_COLLATE=English_United States.1252 
[2] LC_CTYPE=English_United States.1252   
[3] LC_MONETARY=English_United States.1252
[4] LC_NUMERIC=C                          
[5] LC_TIME=English_United States.1252    

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
 [1] latticeExtra_0.6-28   RColorBrewer_1.1-2    lattice_0.20-38      
 [4] htmlwidgets_1.3       ndtv_0.13.0           animation_2.6        
 [7] statnet_2019.6        tsna_0.3.0            sna_2.4              
[10] statnet.common_4.3.0  ergm.count_3.4.0      tergm_3.6.1          
[13] networkDynamic_0.10.0 ergm_3.10.4           network_1.15         
[16] knitr_1.23           

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.1        pillar_1.4.1      compiler_3.6.0   
 [4] DEoptimR_1.0-8    tools_3.6.0       base64_2.0       
 [7] digest_0.6.19     jsonlite_1.6      nlme_3.1-140     
[10] evaluate_0.14     tibble_2.1.3      pkgconfig_2.0.2  
[13] rlang_0.3.4       Matrix_1.2-17     yaml_2.2.0       
[16] parallel_3.6.0    xfun_0.7          coda_0.19-2      
[19] dplyr_0.8.1       stringr_1.4.0     askpass_1.1      
[22] trust_0.1-7       grid_3.6.0        tidyselect_0.2.5 
[25] glue_1.3.1        robustbase_0.93-5 R6_2.4.0         
[28] rmarkdown_1.13    purrr_0.3.2       magrittr_1.5     
[31] htmltools_0.3.6   MASS_7.3-51.4     assertthat_0.2.1 
[34] lpSolve_5.6.13.1  stringi_1.4.3     openssl_1.4      
[37] crayon_1.3.4     

Set seed for simulations – this is not necessary, but it ensures that we all get the same results (if we execute the same commands in the same order).

set.seed(0)

Types of temporal network data

Temporal network information can be collected in different ways, leading to several different types of data:

• Discrete time network census panel data (with 2 or more panels)
• Cross-sectional network census data with information on tie duration
• Cross-sectional sampled network data with tie duration information
• Continuous time network event data

Each type typically requires different methods for analysis. We will cover the first two types of data in this tutorial (though TERGMs can handle the first three).

Overview of modeling frameworks for temporal network data

There are three general modeling frameworks for temporal network data analysis in the social networks field.

• Stochastic Actor Oriented Models (SAOM)–implemented in the RSiena package written by Tom A.B. Snijders and colleagues. SAOMs model the coevolution of nodal attributes and tie dynamics. They are formulated as continuous time models, but typically are estimated on discrete time data.

• Relational Event Models–implemented in the statnet package relevent written by Carter Butts and colleagues. These are continuous time models for node and tie dynamics, so they can assume dyad independence, and exploit the methodology of logistic regression.

• Temporal Exponential family Random Graph Models (TERGMS)–This is a large family of models that extend ERGMs for dynamic networks, and include the specific subclass of Separable TERGMs that are implemented in the statnet software in the tergm package. TERGMs represent tie dynamics only, and are typically formulated in discrete time.

This workshop will focus on TERGMs. And we’ll start with a quick overview of the utilities for data storage, descriptive stats and visualization of temporal network data before moving on to the statistical modeling.

All temporal network models can be thought of as a form of “agent-based” model, in that they can be used for stochastic simulation of a population of “agents” interacting over time. However, the statistical network modeling frameworks are distinguished by (1) a foundation based on principled statistical estimation of the model parameters that allows for hypothesis testing and inference at the model building stage; and (2) reproduction of the full joint distribution of observed network configurations that are represented by the terms in the model.

networkDynamic: storing and working with temporal network data

Temporal network data in statnet are stored as networkDynamic class objects–but these also have the class network. They can thus be edited or queried using standard functions from the network package, or using additional functions tailored specifically to the case of dynamic networks in the package networkDynamic.

Let us consider the 3 panels of the Sampson monastery data:

data(samplk)
ls()

[1] "samplk1" "samplk2" "samplk3" "tb"     

To create a dynamic network object from the panels, we need to combine them into a list:

samplist <- list(samplk1,samplk2,samplk3)
sampdyn <- networkDynamic(network.list = samplist)

Neither start or onsets specified, assuming start=0
Onsets and termini not specified, assuming each network in network.list should have a discrete spell of length 1
Argument base.net not specified, using first element of network.list instead
Created net.obs.period to describe network
 Network observation period info:
  Number of observation spells: 1 
  Maximal time range observed: 0 until 3 
  Temporal mode: discrete 
  Time unit: step 
  Suggested time increment: 1 

Note that, by default, the starting time is set to 0, so the first network will have this time stamp. You can change that behavior by specifying a different value, for example: sampdyn <- networkDynamic(network.list = samplist, start=1).

The networkDynamic function is able to convert a wide array of temporal network data types into the networkDynamic objects. For more information, see ?networkDynamic.

As with network objects you can get a quick summary of a networkDynamic object by typing the object name:

sampdyn

NetworkDynamic properties:
  distinct change times: 4 
  maximal time range: 0 until  3 

Includes optional net.obs.period attribute:
 Network observation period info:
  Number of observation spells: 1 
  Maximal time range observed: 0 until 3 
  Temporal mode: discrete 
  Time unit: step 
  Suggested time increment: 1 

 Network attributes:
  vertices = 18 
  directed = TRUE 
  hyper = FALSE 
  loops = FALSE 
  multiple = FALSE 
  bipartite = FALSE 
  net.obs.period: (not shown)
  total edges= 88 
    missing edges= 0 
    non-missing edges= 88 

 Vertex attribute names: 
    active cloisterville group vertex.names 

 Edge attribute names: 
    active 

Note that the description indicates 4 timepoints (0 through 3). This is an artifact of representing discrete time. If you look at the final network, you’ll see it is empty:

network.extract(sampdyn, at = 3) # empty
network.extract(sampdyn, at = 0) # the first network, as expected

The discrete time representation means that specifying at=0 is equivalent to specifying the semi-open interval [0-1) with onset = 0, terminus = 1.

# Equivalent statements
#network.extract(sampdyn, at = 0) # first network
#network.extract(sampdyn, onset = 0, terminus=1) # same thing

The networkDynamic package contains many utilities for modifying, extracting and querying networkDynamic objects. For example, the network.extract function above extracts one or more networks from the series, while the network.collapse function extracts one or more networks and collapses them into a single network object.

You can get a good overview of the wide range of functionality in this package by reading the package vignette:

vignette("networkDynamic")

As one small example, we’ll just make a quick plot of the three network panels, using the network.extract function (this also extracts one or more networks, but preserves the time info if present) and the network.

par(mfrow = c(2,2), oma=c(1,1,1,1), mar=c(4,1,1,1))
plot(network.extract(sampdyn, at = 0), main = "Time 1", 
     displaylabels = T, label.cex = 0.6, vertex.cex = 2, pad = 0.5)
plot(network.extract(sampdyn, at = 1), main = "Time2", 
     displaylabels = T, label.cex = 0.6, vertex.cex = 2, pad = 0.5)
plot(network.extract(sampdyn, at = 2), main = "Time3", 
     displaylabels = T, label.cex = 0.6, vertex.cex = 2, pad = 0.5)
plot(sampdyn, main = "Collapsed", 
     displaylabels = T, label.cex = 0.6, vertex.cex = 2, pad = 0.5)

Not the most helpful display if you’re seeking some insight into the temporal changes in the friendships among the monks at the Sampson Monastery. The next packages we’ll review are designed to facilitate that task.

tsna: a package for descriptive temporal network analysis

As the name suggests, this package generalizes the sna package functionality for temporal network data. It provides utilities for calculating:

• Static social network analysis metrics at multiple time points via the tSnaStats function

tSnaStats(sampdyn,"degree") # Changes in degree centrality

Time Series:
Start = 0 
End = 3 
Frequency = 1 
  John Bosco Gregory Basil Peter Bonaventure Berthold Mark Victor Ambrose
0         12      10     5     6           8        4    7      7       5
1         11      12     4     8          10        5    6      5       4
2          7       9     7     7           9        5    8      5       7
3         NA      NA    NA    NA          NA       NA   NA     NA      NA
  Romauld Louis Winfrid Amand Hugh Boniface Albert Elias Simplicius
0       4     5       4     5   10        4      5     4          5
1       4     5       7     5    6        6      5     5          6
2       4     5       9     5    5        5      4     5          6
3      NA    NA      NA    NA   NA       NA     NA    NA         NA

• Static ergm term statistics at multiple time points via the tErgmStatsfunction

tErgmStats(sampdyn, "~ edges+triangle") # Notice the increase in triangles

Time Series:
Start = 0 
End = 3 
Frequency = 1 
  edges triangle
0    55       31
1    57       56
2    56       62
3     0        0

• Temporal paths through networks via the tPath, tReach and plotPaths functions

Temporal paths are unique to dynamic networks. Similar to a path in a static network, they trace out the sequence and distances of nodes in a networkDynamic object reachable from an initial node, following paths constrained by edge timing (and direction, if the network is directed). A temporal path can only traverse active nodes and edges, so the onset times of successive elements must be greater than or equal than those of the previous.

Below, we compute and plot the forward reachable set (FRS) from node 1 (v1) in the Sampson networks (the vertices that node 1 can reach).

# who is v1?
get.vertex.attribute(sampdyn, "vertex.names")[1]

[1] "John Bosco"

# who is in v1's FRS?
tp <- tPath(sampdyn, v=1, direction = 'fwd')
print(tp)

$tdist
 [1] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

$previous
 [1]  0  3  1  5  1  4  2  7  6  4  5 14  5  1 14  7  3 13

$gsteps
 [1] 0 2 1 2 1 3 3 4 4 3 2 2 2 1 2 4 2 3

$start
[1] 0

$end
[1] Inf

$direction
[1] "fwd"

$type
[1] "earliest.arrive"

attr(,"class")
[1] "tPath" "list" 

par(mfrow=c(1,2))
coords <- plot(tp, main="Forward Reachable Set from v1", cex.main=.8)
plotPaths(sampdyn, tp, 
          coord = coords,
          main = "Overlaid on collapsed Network",
          label.cex=.8, cex.main=.8)

Because all of the vertices are in one component in the first graph, node 1 (Bosco) is able to reach every other vertex in the first spell; the first network alone determines the temporal path.
A forward tpath traces the earliest arriving path, so in this case, tp$tdist = 0 for all vertices, because they can all be reached in the first spell, tp$gsteps is simply the geodesic from Bonaventure to each other vertex in the first network, and plotting tp shows those geodesics.

And note that one can also construct a “backwards reachable set” for Bosco – the set of all vertices that can reach him.

Forward and backward reachable sets are key metrics for understanding diffusion processes over networks. The tsna package vignette demonstrates many more interesting features of temporal paths in dynamic networks with multiple components that change over time.

• Duration summaries for node and tie dynamics

Durations are another feature unique to temporal networks, representing the lenth of time that vertices and edges are active. The tsna package currently contains several functions for reporting on active durations. Most of these functions have options for varying the units of aggregation or analysis. For example, the edge measures can be adjusted from focusing on an ‘edge’ (aggregating across all of the activity spells associated with that edge) or a ‘spell’ (a single interval of contiguous time during which an edge is activated).

table(edgeDuration(sampdyn, mode = 'duration', subject = 'spells')) # Note bimodality

 1  2  3 
48 15 30 

ndtv: Visualizing dynamic networks

The Network Dynamic Temporal Visualization (ndtv) package provides tools for visualizing changes in network structure and attributes over time.

ndtv has its own workshop and tutorial, with materials online at the ndtv GitHub repo

The package includes an example network data set named short.stergm.sim which is the output of a toy model based on Padgett’s Florentine Family Business Ties dataset simulated using the tergm package. (We’ll use tergm package to construct this simulated dynamic network in a later section.)

library(ndtv)
data(short.stergm.sim)

render.d3movie(short.stergm.sim, 
               plot.par=list(displaylabels=T))

render.d3movie(short.stergm.sim,
               plot.par=list(displaylabels=T),
               output.mode = 'htmlWidget') # using htmlwidgets package here

slice parameters:
  start:0
  end:25
  interval:1
  aggregate.dur:1
  rule:latest

proximity.timeline(short.stergm.sim,default.dist = 6,
          mode = 'sammon',labels.at = 17,vertex.cex = 4)

An Introduction to STERGMs (non-technical)

Where an ERGM provides a single model for the prevalence of ties in a cross-sectional single network, STERGMs posit two models for the tie dynamics in a network over time:

• one for tie formation, and

• one for tie dissolution.

Both models are ERGMs, and they are “separable” in the sense that they assume formation is independent of dissolution within time step, and Markov dependent between steps. This allows the factors that influence formation to be different than those that influence dissolution, so specifying a STERGM requires two formulas instead of one.

Allowing the formation model to be different than the dissolution model is useful in many contexts. Take friendships. It seems intuitive that the factors influencing who becomes friends with whom, out of the set of people who aren’t already) are likely to be different than the factors affecting who stops liking whom, out of the set of people currently in relationships). Any reasonable model of formation would need to include a variety of homophily parameters (mixing on age, for example). Friendship dissolution may or may not. (Conditional on being friends, does a difference in age affect the probability that your friendship ends? Perhaps, but probably not as fundametally or as strongly as for formation).

In thinking about how STERGMs work, and the difference between the prevalence of ties in the cross section, and the incidence of ties in a temporal data set, it can be helpful to think of the approximation commonly used in epidemiology to consider basic disease dynamics:

\[ prevalence = incidence \times duration \]

To translate this to dynamic networks: the expression above says that the number of ties present in the cross-section (prevalence) is a function of the rate at which they form (incidence) and the rate at which they dissolve (1/duration). The faster ties form, the more ties will be present in the cross-section, and the more slowly they form, the fewer will be present. The slower ties dissolve, the longer the duration, the more ties will be present in the cross-section, and the reverse (faster/shorter/fewer) again holds. The cross-sectional prevalence of ties is determined by both processes.

This same concept extends to other network configurations, for example triangles—the faster that triangles are formed and the slower that they’re dissolved, the more that will be present in the cross-section.

ERGMs are models for the prevalence of ties; they do not represent the implicit rates of incidence and duration, they simply model the net result.

STERGMs are models that explicitly represent the incidence and duration processes.

The two equations governing a STERGM are commonly called the formation equation and the dissolution (or persistence) equation, respectively:

These parallel the traditional cross-sectional ERGM. We have simply:

• added a time index to the tie values

• added in a new conditional. In the formation equation, the expression is conditional on the tie not existing at the prior time step, and in the dissolution equation it is conditional on the tie existing; and

• defined two \(\theta\) and \(g\) vectors instead of just one. Now there are \(\theta^+\) and \(g^+(y)_{ij}\), reflecting the coefficients and statistics in the formation model, and \(\theta^-\) and \(g^-(y)_{ij}\) reflecting those in the dissolution model.

Note that for the dissolution model, the \(P(Y_{ij,t+1} = 1)\) is in the numerator and the \(P(Y_{ij,t+1} = 0)\) is in the denominator. This makes the mathematics parallel to that of the formation model; however, it also means the “dissolution” model is actually a “persistence” model. The probability of dissolution is simply 1 - persistence, so the model can be interpreted in both ways.

Finally, remember that the s in stergm stands for separable; this refers to assumption that relational formation and dissolution occur independently of one another within a time step. We explore how this works and its implications in the examples below. But effectively, at each timestep, we are sending off all empty dyads to be considered for formation (conditioning on the ties that exist), and sending off all the paired dyads to be considered for dissolution (conditioning on the empty dyads), and these two processes proceed independently.

An Introduction to STERGMs (more formal)

To understand STERGMs formally, we first review the ERGM framework for cross-sectional or static networks. Let \(\mathbb{Y}\subseteq \{1,\dotsc,n\}^2\) be the set of potential relations (dyads) among \(n\) nodes, ordered for directed networks and unordered for undirected. We can represent a network \(\mathbf{y}\) as a set of ties, with the set of possible sets of ties, \(\mathcal{Y}\subseteq 2^\mathbb{Y}\), being the sample space: \(\mathbf{y} \in \mathcal{Y}\). Let \(\mathbf{y}_{ij}\) be 1 if \((i,j)\in\mathbf{y}\) — a relation of interest exists from \(i\) to \(j\) — and 0 otherwise.

The network also has an associated covariate array \(\mathbf{X}\) containing attributes of the nodes, the dyads, or both. An exponential-family random graph model (ERGM) represents the pmf of \(\mathbf{Y}\) as a function of a \(p\)-vector of network statistics \(g(\mathbf{Y},\mathbf{X})\), with parameters \(\theta \in \mathbb{R}^p\), as follows:

where the normalizing constant \(k(\theta, \mathbf{X}, \mathcal{Y}) = \sum\limits_{\mathbf{y}' \in \mathcal{Y}}\exp\left\{\theta\cdot g(\mathbf{y}', \mathbf{X})\right\}\) is a summation over the space of possible networks on \(n\) nodes, \(\mathcal{Y}\). Where \(\mathcal{Y}\) and \(\mathbf{X}\) are held constant, as in a typical cross-sectional model, they may be suppressed in the notation. Here, on the other hand, the dependence on \(\mathcal{Y}\) and \(\mathbf{X}\) is made explicit.

In modeling the transition from a network \(\mathbf{Y}^{t}\) at time \(t\) to a network \(\mathbf{Y}^{t+1}\) at time \(t+1\), the separable temporal ERGM assumes that the formation and dissolution of ties occur independently from each other within each time step, with each half of the process modeled as an ERGM. For two networks (sets of ties) \(\mathbf{y},\mathbf{y}'\in \mathcal{Y}\), let \(\mathbf{y} \supseteq \mathbf{y}'\) if any tie present in \(\mathbf{y}'\) is also present in \(\mathbf{y}\). Define \(\mathcal{Y}^+(\mathbf{y}) = \{\mathbf{y}'\in\mathcal{Y}:\mathbf{y}'\supseteq\mathbf{y}\}\), the networks that can be constructed by forming ties in \(\mathbf{y}\); and \(\mathcal{Y}^-(\mathbf{y}) = \{\mathbf{y}'\in\mathcal{Y}:\mathbf{y}'\subseteq\mathbf{y}\}\), the networks that can be constructed dissolving ties in \(\mathbf{y}\).

Given \(\mathbf{y}^{t}\), a formation network \(\mathbf{Y}^+\) is generated from an ERGM controlled by a \(p\)-vector of formation parameters \(\theta^+\) and formation statistics \(g^+(\mathbf{y}^+, \mathbf{X})\), conditional on only adding ties:

A dissolution network \(\mathbf{Y}^-\) is simultaneously generated from an ERGM controlled by a (possibly different) \(q\)-vector of dissolution parameters \(\theta^-\) and corresponding statistics \(g^-(\mathbf{y}^-, \mathbf{X})\), conditional on only dissolving ties from \(\mathbf{y}^{t}\), as follows:

The cross-sectional network at time \(t+1\) is then constructed by applying the changes in \(\mathbf{Y}^+\) and \(\mathbf{Y}^-\) to \(\mathbf{y}^{t}\), as follows: \(\mathbf{Y}^{t+1} = \mathbf{Y}^{t}\cup (\mathbf{Y}^+ - \mathbf{Y}^{t}) \, - \, ( \mathbf{Y}^{t} - \mathbf{Y}^-).\) which simplifies to either: \(\mathbf{Y}^{t+1} = \mathbf{Y}^+ - (\mathbf{Y}^{t} - \mathbf{Y}^-)\) \(\mathbf{Y}^{t+1} = \mathbf{Y}^-\cup (\mathbf{Y}^+ - \mathbf{Y}^{t})\)

Visually, we can sum this up as:

stergm Model specification and syntax

In statnet, an ERGM is specified using R’s formula notation:

        ergm(my.network ~ edges + gwesp(0, fixed=T))     #do not run this

For a call to stergm, there is one dynamic network, with two formulas. These are now passed as three separate arguments: the network(s) (argument nw), the formation formula (argument formation), and the dissolution formula (argument dissolution).
The latter two take the form of a one-sided formula. E.g.:

        stergm(my.network,                               #do not run this
            formation =  ~ edges + nodefactor('age'),
            dissolution =  ~ edges + gwesp(0, fixed=T),
            estimate =  `insert method`
        )

In addition to the two equations, one more specification is required for a stergm as shown above: the argument estimate, which controls the estimation method. This is because there are different types of temporal network data, and each requires a different type of estimation.

The two options are CMLE (conditional maximum likelihood estimation) and EGMME (equilibrium generalized method of moments estimation). When fitting a model to network panel data, use estimate = "CMLE"; and when fitting a model to a single cross-sectional network with duration information, use estimate = "EGMME".

There are many other features of a call to stergm that will be important; we illustrate them in one or more examples below.

CMLE: Network panel data (Repeated network cross-sections over time)

The canonical form of data for modeling dynamic network processes consists of observations of a complete network at two or more points in time on the same node set. Many classic temporal network studies were of this type, and this remains a gold standard for temporal network data collection.

Let us return to the Sampson monastery data, and consider models for the formation and dissolution of ties across the 3 observed time points.

This is a directed network, so it would be natural to consider mutuality as a predictor of a directed tie. From the descriptive statistics we saw that the number of triangles increased over time. Given directedness, it would be natural to examine the role of cyclical (non-hierarchical) vs. transitive (hierarchical) triangles in the network. The robust way to model this in ERGMs is with the terms transitiveties and cyclicalties (the number of ties \(i{\rightarrow}j\) that have at least one two-path from \(i{\rightarrow}j\) and from \(j{\rightarrow}i\), respectively).

In this case, we will specify the same terms in both the formation and dissolution model. That is not necessary in stergms, we could just as well specify a simple homogenous Bernoulli model for dissolution (all ties are equally likely to dissolve at each time step). Having the same terms also does not mean we have the “same model” for formation and dissolution. We might see different terms are signficant in the formation model than in the dissolution model; or that the coefficient estimates differ in sign or magnitude.

In the following code, we pass five arguments: the network list that we want to model (samplist), the formation formula, the dissolution formula, the fitting algorithm (CMLE since we have a network panel), and the list of time slices we wish to model:

samp.fit <- stergm(samplist,
    formation =  ~edges+mutual+cyclicalties+transitiveties,
    dissolution = ~edges+mutual+cyclicalties+transitiveties,
    estimate = "CMLE",
    times = c(1:3)
    )

Fitting formation:  

Starting maximum likelihood estimation via MCMLE:
Iteration 1 of at most 20:  

 = = = = = = Lots of output snipped = = = = = = 

This model was fit using MCMC.  To examine model diagnostics and check for degeneracy, use the mcmc.diagnostics() function.

And the results:

summary(samp.fit)

==============================
Summary of formation model fit 
==============================

Formula:   ~edges + mutual + cyclicalties + transitiveties

Iterations:  2 out of 20 

Monte Carlo MLE Results:
               Estimate Std. Error MCMC % z value Pr(>|z|)    
edges           -3.4623     0.3412      0 -10.147   <1e-04 ***
mutual           2.0444     0.4059      0   5.037   <1e-04 ***
cyclicalties    -0.1432     0.1970      0  -0.727   0.4671    
transitiveties   0.3975     0.2353      0   1.689   0.0912 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

     Null Deviance: 693.1  on 500  degrees of freedom
 Residual Deviance: 240.0  on 496  degrees of freedom
 
AIC: 248    BIC: 264.9    (Smaller is better.) 

================================
Summary of dissolution model fit
================================

Formula:   ~edges + mutual + cyclicalties + transitiveties

Iterations:  2 out of 20 

Monte Carlo MLE Results:
               Estimate Std. Error MCMC % z value Pr(>|z|)  
edges            0.2080     0.2990      0   0.696   0.4867  
mutual           0.7987     0.5042      0   1.584   0.1131  
cyclicalties    -0.1886     0.2455      0  -0.768   0.4424  
transitiveties   0.5051     0.2794      0   1.808   0.0706 .
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

     Null Deviance: 155.3  on 112  degrees of freedom
 Residual Deviance: 136.8  on 108  degrees of freedom
 
AIC: 144.8    BIC: 155.6    (Smaller is better.) 

So, all else equal, a relationship is much more likely to form if it will close a mutual pair; the conditional log-odds are increased by 2.04 which translates to an increase in the conditional odds of 7.72 (note that the change statistic can equal only \(0\) or \(1\) in this case).

The reciprocity effect on persistence is also positive, but less strong and not statistically signifiant; the point estimate is an increase of 0.8 in the conditional log-odds of persistence, a bit more than double the conditional odds.

The coefficients for transitive triads (hierarchical) are positive and borderline significant for both formation and persistence. The coefficients for cyclical triads (egalitarian) are negative for both models, but not significant. Keep in mind that this is a small sparse network, so the amount of information available for testing these effects is minimal (especially for the dissolution model). Overall, the results suggest that hierarchy plays a role in tie formation and persistence in this monastery, and there is a weak but consistent anti-egalitarianism dynamic.

To include only a subset of the panels from a network series, you can alter the times argument:

samp.fit.2 <- stergm(samplist,
  formation =  ~edges+mutual+cyclicalties+transitiveties,
    dissolution = ~edges+mutual+cyclicalties+transitiveties,
    estimate = "CMLE",
  times = 1:2
    )

 = = = = = = Lots of output snipped again = = = = = = 

summary(samp.fit.2)

==============================
Summary of formation model fit 
==============================

Formula:   ~edges + mutual + cyclicalties + transitiveties

Iterations:  2 out of 20 

Monte Carlo MLE Results:
               Estimate Std. Error MCMC % z value Pr(>|z|)    
edges           -3.3327     0.4605      0  -7.238   <1e-04 ***
mutual           2.2346     0.5654      0   3.952   <1e-04 ***
cyclicalties    -0.3089     0.2921      0  -1.057    0.290    
transitiveties   0.5084     0.3349      0   1.518    0.129    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

     Null Deviance: 348.0  on 251  degrees of freedom
 Residual Deviance: 131.6  on 247  degrees of freedom
 
AIC: 139.6    BIC: 153.7    (Smaller is better.) 

================================
Summary of dissolution model fit
================================

Formula:   ~edges + mutual + cyclicalties + transitiveties

Iterations:  2 out of 20 

Monte Carlo MLE Results:
               Estimate Std. Error MCMC % z value Pr(>|z|)  
edges            0.1013     0.4213      0   0.241   0.8099  
mutual           1.2956     0.7464      0   1.736   0.0826 .
cyclicalties    -0.3997     0.4235      0  -0.944   0.3452  
transitiveties   0.4037     0.4326      0   0.933   0.3508  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

     Null Deviance: 76.25  on 55  degrees of freedom
 Residual Deviance: 68.34  on 51  degrees of freedom
 
AIC: 76.34    BIC: 84.37    (Smaller is better.) 

How do these coefficients compare? How about the standard errors?

EGMME: Cross sectional network data with tie duration information

Now, imagine a different scenario in which we have observed a single cross-sectional network, with retrospective information collected on the duration of ties. This kind of data can be collected in a simple one-time survey, so it is much easier to obtain, and in many contexts it will the only feasible type of network data that can be collected.

How can we estimate the formation and dissolution models with a single cross-sectional network? By making an equilibrium assumption.

If we assume that the process is in equilibrim, then we can rely on the fact that, at equilibrium, \(prevalence = incidence \times duration\).
Our cross-sectional network allows us to observe the \(prevalence\) of ties, and the retrospective information collected allow us to estimate the \(duration\), so we can use stergm to solve for the \(incidence\) in the formation model that would be consistent with this prevalence and duration.

The duration information required here is obtained by asking respondents to report the duration of relationships. In the simple case of a time-constant dissolution hazard, the mean duration is the estimate we need to calculate the dissolution edge coefficient. This estimate can come from the same cross-sectional study, or, if necessary, from a completely different data source.

We will demonstrate the latter by creating a toy dataset, based on Padgett’s cross-sectional flobusiness network, using a fictional mean tie duration estimate of 10 time steps, and assuming a homogeneous dissolution process (that is, every existing tie has the same probability of dissolving, at each step). With real data we could model heterogeneous durations as we did above, but in this simple case will suffice to demonstrate the basic modeling tasks.

• First, we specify the formation and dissolution models.

We begin with a model that sets the density (edges) and includes a term to test for possible triad formation bias (gwesp):

formation = ~ edges + gwesp(alpha=0,fixed = T)

Note that with the decay parameter alpha set to \(0\) the gwesp term only counts the first triad closed by a tie so the coefficient will represent the log-odds of forming a tie given that it creates at least one “shared-parter” triad.

For our dissolution model, we will assume a simple homogenous process, which corresponds to a model with only an edges term in it. The coefficient on this term represents the conditional log-odds of tie persistence at each step, and the inverse is the mean log duration.

In this case, however, we already know the mean duration, which means we know the rate of persistence/dissolution, so we don’t need to estimate it.

A known coefficient in the statistical modeling context is called an “offset”.

In STERGM notation we specify this as:

dissolution = ~offset(edges)

• Second, we calculate the offset: theta.diss.

Our dissolution model is applied to the set of ties that exist at each step, as reflected in the conditional present in both the numerator and denominator:

\[ \ln{\frac{P(Y_{ij,t+1} = 1 \mid Y_{ij,t} = 1)}{P(Y_{ij,t+1} = 0\mid Y_{ij,t} = 1)}} = \theta'\delta(g(y))_{ij} \]

The numerator represents the case when a tie persists to the next step, and the denominator represents the case when it dissolves.

The one term in our \(\delta(g(y))_{ij}\) vector is the change statistic for the network edge count, which equals one for all \(i,j\).

Let the probability of persistence from one time step to the next for actor pair \(i,j\) be \(p_{ij}\), and the probability of dissolution be \(q_{ij} = 1-p_{ij}\). Our dissolution model is homogenous, so we further define \(p_{ij} = p\) for all \(i,j\). Then:

This is a discrete memoryless process, so the durations have a geometric distribution; and for the geometric distribution, the mean duration is equal to the reciprocal of the dissolution probability. Symbolizing mean relational duration as \(d\), we have \(d = \frac{1}{q}\), and thus:

So, for our dissolution model, theta.diss = \(\ln{(10-1)}\) = \(\ln{9}\) (\(\approx 2.2\)):

theta.diss <- log(9)

In short, we can calculate the dissolution coefficient using a (rather simple) closed form solution in this case. And in general, we can do this for all dyadic-dependent terms in a dissolution model.

• Third, we specify our equilibrium target statistics

This is the second difference in the EGMME specification.

We specify this with a one-sided formula listing the statistics that will be used as equilbrium targets.

The targets can be one of three things: the statistics in the formation model, the statistics in the dissolution model, or a different set of statistics altogether. If the formula is identical to either the formation or dissolution model, the user can simply pass the string "formation" or "dissolution", respectively.

Note: If one is specifying targets = "formation", then the dissolution coefficient(s) should be specified with an offset, and vice versa.

Typically, the target statistic values are drawn from the observed cross-sectional network that is the focus of analysis. In this case, they are the values of the terms returned by a call to the summary function.

For stergms, as for ergms, however, the target values do not have to come from a specific observed network – they can be derived as counterfactuals, or from multiple data sets, as long as they are internally consistent (for example, the number of ties for a nodefactor level cannot exceed the total number of ties in the network). In this case, we would pass the values of the target statistics to the model with the argument target.stats. If that argument appears in the model specification, it overrides the values in the observed network data.

Here, we will use the formation model to define the statistics, and take the values from our observed cross-sectional network.

targets = "formation"

• Finally, we estimate the formation model, conditional on the dissolution model.

We put this all together in the call to stergm, adding in one additional control argument to monitor model convergence: plotting the progress of the coefficient estimates and the simulated sufficient statistics in real time. When one is using a GUI tool like RStudio, it helps to output this plotting to a separate window, which we do below with the function X11 (and then turn off that window again with dev.off()):

data(florentine)
X11()
stergm.fit.1 <- stergm(flobusiness,
    formation =  ~edges+gwesp(0,fixed = T),
    dissolution = ~offset(edges),
    targets = "formation",
    offset.coef.diss = theta.diss,
    estimate = "EGMME",
    control = control.stergm(SA.plot.progress = TRUE)
    )
dev.off()

Iteration 1 of at most 20:  

 = = = = = = Lots of output snipped = = = = = = 

 = = Phase 3: Simulate from the fit and estimate standard errors. =  = 

In the X11 window, you should see the MCMC traceplots showing the progress of the coefficient estimates (the f. plots) and the corresponding values of the target statistics.

The real-time feedback suggests that the fitting went well, but let’s double-check:

mcmc.diagnostics(stergm.fit.1)

 = = = = = = = = = = = = = = = = = = = = = = = = =  = 
EGMME diagnostics
 = = = = = = = = = = = = = = = = = = = = = = = = =  = 
 = = = = = = Lots of output snipped = = = = = = 

These look pretty good. The serial correlation evident in the traceplots could be reduced by modifying some of stergm’s control parameters. You can experiment with increasing the mcmc.interval to see how that works. But the level of serial correlation here is not enough to cause concern.

So, we’ll move on to explore the fit object to see what we have:

stergm.fit.1

Formation Coefficients:
        edges  gwesp.fixed.0  
       -6.608          2.465  
Dissolution Coefficients:
offset(edges)  
        2.197  

names(stergm.fit.1)

 [1] "network"         "formation"       "dissolution"    
 [4] "targets"         "target.stats"    "estimate"       
 [7] "covar"           "opt.history"     "sample"         
[10] "sample.obs"      "control"         "reference"      
[13] "mc.se"           "constraints"     "formation.fit"  
[16] "dissolution.fit"

summary(stergm.fit.1)

Warning: This object was fit with 'ergm' version 3.9.4 or earlier.
Summarizing it with version 3.10 or later may return incorrect results or
fail.

Warning: This object was fit with 'ergm' version 3.9.4 or earlier.
Summarizing it with version 3.10 or later may return incorrect results or
fail.

==============================
Summary of formation model fit 
==============================

Formula:   ~edges + gwesp(0, fixed = T)

Iterations:  NA 

Equilibrium Generalized Method of Moments Results:
              Estimate Std. Error MCMC % z value Pr(>|z|)    
edges           -6.608      1.562      0  -4.231   <1e-04 ***
gwesp.fixed.0    2.465      1.884      0   1.308    0.191    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


================================
Summary of dissolution model fit
================================

Formula:   ~offset(edges)

Iterations:  NA 

Equilibrium Generalized Method of Moments Results:
              Estimate Std. Error MCMC % z value Pr(>|z|)    
offset(edges)    2.197      0.000      0     Inf   <1e-04 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


 The following terms are fixed by offset and are not estimated:
  offset(edges) 

We have now obtained estimates for the coefficients of a formation model that, conditional on the given dissolution model (and it’s offsets), yields simulated targets over time that match (in expectation) those observed in the cross-sectional flobusiness network.

From this fitted model we can now simulate new dynamic networks (that is, entire network panels over time) that have the desired cross-sectional structure and mean relational duration. This is a very useful tool – essentially an agent-based simulation that is guaranteed to reproduce, on average, the full joint distribution of the cross-sectional network statistics and partnership durations in the model that we observed in our data.

stergm.sim.1 <- simulate(stergm.fit.1, nsim = 1, 
    time.slices = 1000)

We can use ndtv to visualize the simulated network time series (we’ll just look at the 1st 25 steps):

wealthsize <- log(get.vertex.attribute(flobusiness, "wealth")) * 2/3
slice.par = list(start = 0, 
               end = 25, 
               interval = 1, 
               aggregate.dur = 1, 
               rule = "any")
compute.animation(stergm.sim.1, slice.par = slice.par)
render.par = list(tween.frames = 5,
                show.time = T,
                show.stats = "~edges+gwesp(0,fixed = T)")
plot.par = list(edge.col = "darkgray",
              displaylabels = T,
              label.cex = .8,
              label.col = "blue",
              vertex.cex = wealthsize)
render.d3movie(stergm.sim.1,
               render.par = render.par,
               plot.par = plot.par,
               output.mode = 'htmlWidget')

How well do the cross-sectional networks within our simulated dynamic network fit the probability distribution implied by our model? We can check by comparing the summary statistics for our observed network to those for our cross-sectional networks. This is easy, since by default, the simulate command for a stergm object not only simulates a dynamic network, but calculates the statistics from the model for each time point. These are stored in attribute(simulated.networkDynamic)$stats.

cbind(model = summary(flobusiness~edges+gwesp(0,fixed = T)),
      obs = colMeans(attributes(stergm.sim.1)$stats))

              model    obs
edges            15 18.136
gwesp.fixed.0    12 14.777

And we can also easily look at a time series and histogram for each statistic:

plot(attributes(stergm.sim.1)$stats)

Or a scatter plot of the two network statistics for each simulated network cross-section, to see how these co-vary for this model:

plot(as.matrix(attributes(stergm.sim.1)$stats))

We should also check to make sure that our mean duration is what we specified (10 time steps).

This time we’ll use a different approach, using as.data.frame, which, when applied to an object of class networkDynamic, generates a timed edgelist. From this we can directly calculate the mean. We’ll show that this produces the same result as the edgeDuration function.

# create dataFrame for direct estimation
stergm.sim.1.df <- as.data.frame(stergm.sim.1)
names(stergm.sim.1.df)

[1] "onset"             "terminus"          "tail"             
[4] "head"              "onset.censored"    "terminus.censored"
[7] "duration"          "edge.id"          

stergm.sim.1.df[1,]

  onset terminus tail head onset.censored terminus.censored duration
1     0       18    3    5           TRUE             FALSE       18
  edge.id
1       1

cbind(directEst = mean(stergm.sim.1.df$duration),
      ndEst = mean(edgeDuration(stergm.sim.1, 
                  mode = 'duration', 
                  subject = 'spells')))

     directEst    ndEst
[1,]  9.827287 9.827287

Although right-censoring is present for some edges in our simulation, with a mean duration of 10 time steps and a simulation 1000 time steps long, the effect of censoring on our observed mean duration should be small, and we see above that it is.

EGMME Approximation with long durations

When the dissolution model offsets are a subset of terms from the formation model – as was the case in the example above – it has been shown that a good set of starting values for the estimation of the formation model are as follows:

1. Fit the terms in the formation model using a static ERGM on the cross-sectional network; then

2. Subtract the values of the dissolution offsets from the estimated coefficients for the corresponding terms in the cross-sectional ERGM.

The result is a vector of parameter values that are a reasonable place to start the MCMC chain for the estimation of the formation model.
This is in fact what the stergm estimation code does by default for this type of model.

In fact … when mean relational duration is very long, this approximation is so good that it may not be necessary to run a STERGM estimation at all.

This turns out to be quite useful, since models with long mean duration are precisely the ones that are the slowest and most difficult to fit using EGMME. That’s because, with long durations, very few ties dissolve at each MCMC step (toggle), so the fitting algorithm gets very little information to update the estimates, and needs many (many) more steps to accumulate that information.

Again, the equation from epidemiology sheds light on why this works :

\[ incidence \approx prevalence \div duration \]

The edge offset for the dissolution/persistence model gives us log-odds that map onto duration. Fitting the cross-sectional ergm gives us log-odds for prevalence. To estimate incidence, we subtract the duration from prevalence, rather than divide, since we are working on a log scale.

It bears repeating that the terms in your dissolution model must be a subset of the terms in your formation model for you to be able to take advantage of this approximation.

To illustrate, let us reconsider Example 2, with a mean relational duration of 100 time steps.

theta.diss.100 <- log(99)

First, we treat the formation process as if it were a stand-alone cross-sectional model, and estimate it using a standard cross-sectional ERGM. (We fit this same formation model in an earlier call to stergm, so will snip the output here):

ergm.fit1 <- ergm(flobusiness ~ edges + gwesp(0, fixed = T))

summary(ergm.fit1)

==========================
Summary of model fit
==========================

Formula:   flobusiness ~ edges + gwesp(0, fixed = T)

Iterations:  2 out of 20 

Monte Carlo MLE Results:
              Estimate Std. Error MCMC % z value Pr(>|z|)    
edges          -3.3757     0.6068      0  -5.563  < 1e-04 ***
gwesp.fixed.0   1.5752     0.5740      0   2.744  0.00606 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

     Null Deviance: 166.36  on 120  degrees of freedom
 Residual Deviance:  78.13  on 118  degrees of freedom
 
AIC: 82.13    BIC: 87.7    (Smaller is better.) 

theta.form <- ergm.fit1$coef 
theta.form

        edges gwesp.fixed.0 
    -3.375664      1.575223 

Next, we subtract the values of the dissolution \(\theta\) from each of the corresponding values in the formation model.

In this example, the dissolution model contains only an edges term, so this coefficient should be subtracted from the starting value for the edges term in the formation model.

theta.form[1] <- theta.form[1] - theta.diss.100

How well does this approximation do in capturing our desired dynamic network properties? First, we can simulate from it:

stergm.sim.2 <- simulate(flobusiness, 
                         formation = ~edges+gwesp(0,fixed = T),
                         dissolution = ~edges, 
                         monitor = "all",
                         coef.form = theta.form, 
                         coef.diss = theta.diss.100,
                         time.slices = 50000)

Then check the results in terms of cross-sectional network structure and mean relational duration.

# first, recovery of the cross sectional statistics in the formation model
cbind(observed = summary(flobusiness~edges+gwesp(0,fixed = T)),
      simulated = colMeans(attributes(stergm.sim.2)$stats))

              observed simulated
edges               15  15.95886
gwesp.fixed.0       12  12.83824

plot(attributes(stergm.sim.2)$stats)

# second, recovery of the tie duration
stergm.sim.dm.2 <- as.data.frame(stergm.sim.2)
mean(stergm.sim.dm.2$duration)

[1] 99.6451